Question: Find the slope and y-intercept of the line that is ${\text{perpendicular}}$ to $\enspace {y = \dfrac{3}{2}x + 1}\enspace$ and passes through the point ${(7, 4)}$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Solution: Lines are considered perpendicular if their slopes are negative reciprocals of each other. The slope of the blue line is ${\dfrac{3}{2}}$ , and its negative reciprocal is ${-\dfrac{2}{3}}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = -\dfrac{2}{3}x + b}\enspace$ We can plug our point, $(7, 4)$ , into this equation to solve for ${b}$ , the y-intercept. $4 = {-\dfrac{2}{3}}(7) + {b}$ $4 = -\dfrac{14}{3} + {b}$ $4 + \dfrac{14}{3} = {b} = \dfrac{26}{3}$ The equation of the perpendicular line is $\enspace {y = -\dfrac{2}{3}x + \dfrac{26}{3}}\enspace$. ${m = -\dfrac{2}{3}, \enspace b = \dfrac{26}{3}}$